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%I #20 Jul 01 2015 19:24:02
%S 2,414,556,3962,4972,6151,6521,8440,8665,13769,13909,15576,16696,
%T 17176,19926,20630,21541,27090,30822,62118,65349,74014,94203,98600,
%U 101231,103058,108333,112332,136036,142714,145588,147150,160730,162366,169137,194681,200837
%N Numbers n such that the sum of the distinct prime factors of prime(n)-1 and prime(n+1)-1 are the same.
%C Although there are more terms than A105403 so far, these numbers are still fairly uncommon.
%C Is this sequence infinite?
%C It would follow from the generalized Bunyakovsky conjecture that, e.g., there are infinitely many primes p such that p+2, p+12, p+14, 6*p^2+84*p+1 and 6*p^2+84*p+145 are all prime, and there are no primes between 6*p^2+84*p+1 and 6*p^2+84*p+145. If so, then the sequence is infinite, because it contains n where prime(n) = 6*p^2+84*p+1, with prime(n)-1 having distinct prime factors 2,3,p,p+14 and prime(n+1) having distinct prime factors 2,3,p+2,p+12. - _Robert Israel_, Jun 30 2015
%e The prime factors of prime(414)-1 are 2,3,5,5,19 and the prime factors of prime(415)-1 are 2,2,2,3,7,17. The sum of the distinct entries in each of these lists is 29.
%p Primes:= select(isprime,[2,seq(2*i+1,i=1..10^6)]):
%p spf:= map(p -> convert(numtheory:-factorset(p-1),`+`), Primes):
%p select(t -> spf[t+1]=spf[t], [$1..nops(Primes)-1]);
%t Select[Range@ 250000, Total[First /@ FactorInteger[Prime@ # - 1]] == Total[First /@ FactorInteger[Prime[# + 1] - 1]] &] (* _Michael De Vlieger_, Jul 01 2015 *)
%o (PARI) spf(n) = {my(f=factor(n)); sum(k=1, #f~, f[k,1]);}
%o lista(nn) = {forprime(p=2, nn, if (spf(p-1)==spf(nextprime(p+1)-1), print1(primepi(p), ", ")););} \\ _Michel Marcus_, Jun 30 2015
%Y Cf. A008472, A105403, A259564.
%K nonn
%O 1,1
%A _Otis Tweneboah_, _Pratik Koirala_, _Eugene Fiorini_, _Nathan Fox_, Jun 30 2015
%E More terms from _Alois P. Heinz_, Jun 30 2015
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